Problem: Select all polynomials that have $(x-2)$ as a factor. Choose all answers that apply: Choose all answers that apply: (Choice A) A $A(x)=x^3+x^2+4$ (Choice B) B $B(x)=x^3-x-6$ (Choice C) C $C(x)=x^4+3x-10$ (Choice D) D $D(x)=x^4-2x^3-2$
Solution: The following statements are equivalent: $(x-2)$ is a factor of $p(x)$ $p(x)$ is divisible by $(x-2)$ The remainder of $\dfrac{p(x)}{x-2}$ is $0$ We can use the polynomial remainder theorem to solve this problem: For a polynomial $p(x)$ and a number $a$, the remainder on division by $x-2$ is $p(a)$. According to the theorem, the remainder when $p(x)$ is divided by $(x-{2})$ is equal to $p({2})$. So to check each polynomial if it has $(x-2)$ as a factor, we need to check if that polynomial's value at ${x=2}$ is zero. $\begin{aligned} A({2})&=16 \\\\ B({2})&=0 \\\\ C({2})&=12 \\\\ D({2})&=-2 \end{aligned}$ In conclusion, the following polynomial has $(x-2)$ as a factor: $B(x)=x^3-x-6$